I have two groups of a vaccinated people. Group A receives only two doses while the second group receives three doses. Over time, people move from Group A to group B by receiving a second dosage. Each day, we have data on the number of critical condition patients, and how many people are within each group (the data is day-wise deterministic).
If the groups were constant, we could have calculated the statistical significance of the critical patients, and determine the benefits of the third dosage.
Assuming a causal link, between the third dose vaccination and the number of critical patients, how do I determine in the case of group transitions, if the third dose is efficient? Are there other tools other than statistical significance? Is there another way to apply statistical significance? How do they validate such data in scientific researches?
Also, the percentages of critical patients are very small for both groups. Does this change the way we calculate?
This question regards real data on COVID-19 vaccination. As you know, there is no decisive constant $n$ incubation period. The rate of people transitioning between groups is random and uncontrolled.
To sum up, what mathematical tools are available for the assertion that the third dose decreases the number of critical patients, assuming a causal link between the two?
As has been mentioned in the comments, this is not quite a mathematical question, and the question lacks specificity. However, without writing off your query with no comment, I would like to point out one major consideration to hopefully help you understand the question you might be trying to ask:
What is statistical significance?
If you are familiar with statistical significance, then you might realize that determining whether or not the results of a study have "statistical significance" is dependent on the researcher's on choice. In case this is in unfamiliar territory, the first thing we would want to formulate is a set of hypotheses.
What is your null hypothesis, and what is your alternative hypothesis? What is the parameter you are measuring? What distribution are we using? These are all questions that should be answered before being able to talk about statistical significance.
Broadly speaking, statistical significance is tied to the likelihood of an experiment result to have been due to random chance, but requires a proper formulation of the problem you are studying in order to properly assess the related probabilities.
In short, the question you are asking seems to not be a question over statistical significance, but one that is more along the lines of a query regarding Experimental Design. This falls outside of the scope of Mathematics.